SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Electrical reactance</title>

</head>

Electrical reactance

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">"Reactance (physics)" redirects here. For other uses, see <a href="Reactance_(disambiguation)" class="mw-redirect mw-disambig" title="Reactance (disambiguation)">Reactance</a>.</div>

In electrical circuits, reactance is the opposition presented to <a href="Alternating_current" title="Alternating current">alternating current</a> by <a href="Inductance" title="Inductance">inductance</a> and <a href="Capacitance" title="Capacitance">capacitance</a>.<a href="#cite_note-veley01-1">[1]</a> It's measured in <a href="Ohm" title="Ohm">Ω (Ohms)</a>.<a href="#cite_note-veley01-1">[1]</a> Along with resistance, it is one of two elements of <a href="Electrical_impedance" title="Electrical impedance">impedance</a>; however, while both elements involve transfer of electrical energy, no <a href="Joule_heating" title="Joule heating">dissipation of electrical energy as heat</a> occurs in reactance; instead, the reactance stores energy until a quarter-cycle later when the energy is returned to the circuit. Greater reactance gives smaller current for the same applied <a href="Voltage" title="Voltage">voltage</a>.

Reactance is used to compute <a href="Amplitude" title="Amplitude">amplitude</a> and <a href="Phase_(waves)" title="Phase (waves)">phase</a> changes of <a href="Sine_wave" title="Sine wave">sinusoidal</a> alternating current going through a circuit element. Like resistance, reactance is measured in <a href="Ohm" title="Ohm">ohms</a>, with positive values indicating inductive reactance and negative indicating capacitive reactance. It is denoted by the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. An ideal <a href="Resistor" title="Resistor">resistor</a> has zero reactance, whereas ideal reactors have no shunt conductance and no series resistance. As <a href="Frequency" title="Frequency">frequency</a> increases, inductive reactance increases and capacitive reactance decreases.

<div class="mw-heading mw-heading2"><h2 id="Comparison_to_resistance">Comparison to resistance</h2></div>

Reactance is similar to resistance in that larger reactance leads to smaller currents for the same applied voltage. Further, a circuit made entirely of elements that have only reactance (and no resistance) can be treated the same way as a circuit made entirely of resistances. These same techniques can also be used to combine elements with reactance with elements with resistance, but <a href="Complex_number" title="Complex number">complex numbers</a> are typically needed. This is treated below in the section on <a href="Electrical_impedance" title="Electrical impedance">impedance</a>.

There are several important differences between reactance and resistance, though. First, reactance changes the phase so that the current through the element is shifted by a quarter of a cycle relative to the phase of the voltage applied across the element. Second, power is not dissipated in a purely reactive element but is stored instead. Third, reactances can be negative so that they can 'cancel' each other out. Finally, the main circuit elements that have reactance (capacitors and inductors) have a frequency dependent reactance, unlike resistors which have the same resistance for all frequencies, at least in the ideal case.

The term reactance was first suggested by French engineer Édouard Hospitalier in L'Industrie Electrique on 10 May 1893. It was officially adopted by the <a href="American_Institute_of_Electrical_Engineers" title="American Institute of Electrical Engineers">American Institute of Electrical Engineers</a> in May 1894.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Capacitive_reactance">Capacitive reactance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Capacitance" title="Capacitance">Capacitance</a></div>

A capacitor consists of two <a href="Electric_conduction" class="mw-redirect" title="Electric conduction">conductors</a> separated by an <a href="Electrical_insulation" class="mw-redirect" title="Electrical insulation">insulator</a>, also known as a <a href="Dielectric" title="Dielectric">dielectric</a>.

Capacitive reactance is an opposition to the change of voltage across an element. Capacitive reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}}</annotation>

</semantics>

</math><img src="./bbe98111b842bc759e8ca808cfc6a23d3b2b0317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle X_{C}}" loading="lazy"> is <a href="Inversely_proportional" class="mw-redirect" title="Inversely proportional">inversely proportional</a> to the signal <a href="Frequency" title="Frequency">frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> (or <a href="Angular_frequency" title="Angular frequency">angular frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">) and the <a href="Capacitance" title="Capacitance">capacitance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.<a href="#cite_note-Irwin-3">[3]</a>

There are two choices in the literature for defining reactance for a capacitor. One is to use a uniform notion of reactance as the imaginary part of impedance, in which case the reactance of a capacitor is the negative number,<a href="#cite_note-Irwin-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-Glisson-5">[5]</a>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}=-{\frac {1}{\omega C}}=-{\frac {1}{2\pi fC}}}</annotation>

</semantics>

</math><img src="./b7998e89e869c9be0ff12c785390a9157d947cfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.642ex; height:5.676ex;" alt="{\displaystyle X_{C}=-{\frac {1}{\omega C}}=-{\frac {1}{2\pi fC}}}" loading="lazy">.</dd></dl>

Another choice is to define capacitive reactance as a positive number,<a href="#cite_note-6">[6]</a><a href="#cite_note-Hughes-7">[7]</a><a href="#cite_note-8">[8]</a>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}}</annotation>

</semantics>

</math><img src="./e33bdd36f50b2832a361a5982bb57c59ee58e6a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.026ex; height:5.676ex;" alt="{\displaystyle X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}}" loading="lazy">.</dd></dl>

In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{c}=-jX_{c}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{c}=-jX_{c}}</annotation>

</semantics>

</math><img src="./36d4ad65ca09b6f5898216cf89894223fe8b0a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.265ex; height:2.509ex;" alt="{\displaystyle Z_{c}=-jX_{c}}" loading="lazy">.

At <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=0}</annotation>

</semantics>

</math><img src="./1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy">, the magnitude of the capacitor's reactance is infinite, behaving like an <a href="https://en.wiktionary.org/wiki/open_circuit" class="extiw external" title="wikt:open circuit">open circuit</a> (preventing any <a href="Electric_current" title="Electric current">current</a> from flowing through the dielectric). As frequency increases, the magnitude of reactance decreases, allowing more current to flow. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> approaches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

</semantics>

</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">, the capacitor's reactance approaches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, behaving like a <a href="Short_circuit" title="Short circuit">short circuit</a>.

The application of a <a href="Direct_current" title="Direct current">DC</a> voltage across a capacitor causes positive <a href="Electrical_charge" class="mw-redirect" title="Electrical charge">charge</a> to accumulate on one side and negative <a href="Electrical_charge" class="mw-redirect" title="Electrical charge">charge</a> to accumulate on the other side; the <a href="Electric_field" title="Electric field">electric field</a> due to the accumulated charge is the source of the opposition to the current. When the <a href="Potential" title="Potential">potential</a> associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply (ideal AC current source), a capacitor will only accumulate a limited amount of charge before the potential difference changes polarity and the charge is returned to the source. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

<div class="mw-heading mw-heading2"><h2 id="Inductive_reactance">Inductive reactance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Inductance" title="Inductance">Inductance</a></div>

Inductive reactance is a property exhibited by an inductor, and inductive reactance exists based on the fact that an electric current produces a magnetic field around it. In the context of an AC circuit (although this concept applies any time current is changing), this magnetic field is constantly changing as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire (counter-EMF), in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field (known as <a href="Lenz's_law" title="Lenz's law">Lenz's law</a>). Hence, inductive reactance is an opposition to the change of current through an element.

For an ideal inductor in an AC circuit, the inhibitive effect on change in current flow results in a delay, or a phase shift, of the alternating current with respect to alternating voltage. Specifically, an ideal inductor (with no resistance) will cause the current to lag the voltage by a quarter cycle, or 90°.

In electric power systems, inductive reactance (and capacitive reactance, however inductive reactance is more common) can limit the power capacity of an AC transmission line, because power is not completely transferred when voltage and current are out-of-phase (detailed above). That is, current will flow for an out-of-phase system, however real power at certain times will not be transferred, because there will be points during which instantaneous current is positive while instantaneous voltage is negative, or vice versa, implying negative power transfer. Hence, real work is not performed when power transfer is "negative". However, current still flows even when a system is out-of-phase, which causes transmission lines to heat up due to current flow. Consequently, transmission lines can only heat up so much (or else they would physically sag too much, due to the heat expanding the metal transmission lines), so transmission line operators have a "ceiling" on the amount of current that can flow through a given line, and excessive inductive reactance can limit the power capacity of a line. Power providers utilize capacitors to shift the phase and minimize the losses, based on usage patterns.

Inductive reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}}</annotation>

</semantics>

</math><img src="./59f14ed86135c464841bbc6fba36cfb3997df2ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.276ex; height:2.509ex;" alt="{\displaystyle X_{L}}" loading="lazy"> is <a href="Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the sinusoidal signal <a href="Frequency" title="Frequency">frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and the <a href="Inductance" title="Inductance">inductance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">, which depends on the physical shape of the inductor:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}=\omega L=2\pi fL}</annotation>

</semantics>

</math><img src="./a01029033173e630d9e2795da17b2b6e79d5bf6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.857ex; height:2.509ex;" alt="{\displaystyle X_{L}=\omega L=2\pi fL}" loading="lazy">.

The average current flowing through an <a href="Inductance" title="Inductance">inductance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> in series with a <a href="Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> AC voltage source of RMS <a href="Amplitude" title="Amplitude">amplitude</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is equal to:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{L}={A \over \omega L}={A \over 2\pi fL}.}</annotation>

</semantics>

</math><img src="./99bc1e96bbe186fb150df0771d843575de109c64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.275ex; height:5.843ex;" alt="{\displaystyle I_{L}={A \over \omega L}={A \over 2\pi fL}.}" loading="lazy"></dd></dl>

Because a <a href="Square_wave_(waveform)" title="Square wave (waveform)">square wave</a> has multiple amplitudes at sinusoidal <a href="Harmonic" title="Harmonic">harmonics</a>, the average current flowing through an <a href="Inductance" title="Inductance">inductance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> in series with a square wave AC voltage source of RMS <a href="Amplitude" title="Amplitude">amplitude</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is equal to:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{L}={A\pi ^{2} \over 8\omega L}={A\pi \over 16fL}}</annotation>

</semantics>

</math><img src="./4894bca448767285f2fa3d9ed33ca9304295a1c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.621ex; height:6.176ex;" alt="{\displaystyle I_{L}={A\pi ^{2} \over 8\omega L}={A\pi \over 16fL}}" loading="lazy"></dd></dl>

making it appear as if the inductive reactance to a square wave was about 19% smaller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{L}={16 \over \pi }fL}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}={16 \over \pi }fL}</annotation>

</semantics>

</math><img src="./dd9a8b5f2673c870960151d50591b77fd676dc3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.397ex; height:5.176ex;" alt="{\displaystyle X_{L}={16 \over \pi }fL}" loading="lazy"> than the reactance to the AC sine wave.

Any conductor of finite dimensions has inductance; the inductance is made larger by the multiple turns in an <a href="Electromagnetic_coil" title="Electromagnetic coil">electromagnetic coil</a>. <a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a> of electromagnetic induction gives the counter-<a href="Electromotive_force" title="Electromotive force">emf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"> (voltage opposing current) due to a rate-of-change of <a href="Magnetic_flux_density" class="mw-redirect" title="Magnetic flux density">magnetic flux density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {B}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {B}}</annotation>

</semantics>

</math><img src="./2b8b91c469e83c445f3c1adfcee7d6a11dd6a051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.247ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {B}}" loading="lazy"> through a current loop.

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}}</annotation>

</semantics>

</math><img src="./e4ff61e87118da0be39e6932bbc467b69d456019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.427ex; height:5.509ex;" alt="{\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}}" loading="lazy"></dd></dl>

For an inductor consisting of a coil with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> loops this gives:

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=-N{d\Phi _{B} \over dt}}</annotation>

</semantics>

</math><img src="./a53a6a439e8dbdba1754e36c92d671e47d0b08e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.491ex; height:5.509ex;" alt="{\displaystyle {\mathcal {E}}=-N{d\Phi _{B} \over dt}}" loading="lazy">.</dd></dl>

The counter-emf is the source of the opposition to current flow. A constant <a href="Direct_current" title="Direct current">direct current</a> has a zero rate-of-change, and sees an inductor as a <a href="Short-circuit" class="mw-redirect" title="Short-circuit">short-circuit</a> (it is typically made from a material with a low <a href="Resistivity" class="mw-redirect" title="Resistivity">resistivity</a>). An <a href="Alternating_current" title="Alternating current">alternating current</a> has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

<div class="mw-heading mw-heading2"><h2 id="Impedance">Impedance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Electrical_impedance" title="Electrical impedance">Electrical impedance</a></div>

Both reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {X}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {X}}</annotation>

</semantics>

</math><img src="./b304d4de2976f610f0bb24237028399681e16a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle {X}}" loading="lazy"> and <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {R}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {R}}</annotation>

</semantics>

</math><img src="./3abe7f424f59ed9925c7f622af7aead87831412b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle {R}}" loading="lazy"> are components of <a href="Electrical_impedance" title="Electrical impedance">impedance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {Z} }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathbf {Z} }}</annotation>

</semantics>

</math><img src="./2da9988aa7959c6e2c145c7cc4d73c152d9ed50b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle {\mathbf {Z} }}" loading="lazy">.

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =R+\mathbf {j} X}</annotation>

</semantics>

</math><img src="./fead8f66a1a2b1e1d001d7bdfbb918f3a76b0e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.133ex; height:2.509ex;" alt="{\displaystyle \mathbf {Z} =R+\mathbf {j} X}" loading="lazy"></dd></dl>

where:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation>

</semantics>

</math><img src="./b776aaf12c2da4b78ca777cb8295c2000bfd51f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }" loading="lazy"> is the complex <a href="Electrical_impedance" title="Electrical impedance">impedance</a>, measured in <a href="Ohm" title="Ohm">ohms</a>;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, measured in ohms. It is the real part of the impedance: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {R={\text{Re}}{(\mathbf {Z} )}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {R={\text{Re}}{(\mathbf {Z} )}}}</annotation>

</semantics>

</math><img src="./b4e40f2419cf510e737d38682fb487a9233f7adb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.049ex; height:2.843ex;" alt="{\displaystyle {R={\text{Re}}{(\mathbf {Z} )}}}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the reactance, measured in ohms. It is the imaginary part of the impedance: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {X={\text{Im}}{(\mathbf {Z} )}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {X={\text{Im}}{(\mathbf {Z} )}}}</annotation>

</semantics>

</math><img src="./6c01889ffcc7a5b1a9f4bd6629459024ab827804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.297ex; height:2.843ex;" alt="{\displaystyle {X={\text{Im}}{(\mathbf {Z} )}}}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} }</annotation>

</semantics>

</math><img src="./ca5d874dbb32b8b33e83ca521f592808387a486e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:0.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} }" loading="lazy"> is the <a href="Imaginary_unit" title="Imaginary unit">square root of negative one</a>, usually represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation>

</semantics>

</math><img src="./dc909dd0c91a68cdb9cda4ae133fa1f82c987c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }" loading="lazy"> in non-electrical formulas. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} }</annotation>

</semantics>

</math><img src="./ca5d874dbb32b8b33e83ca521f592808387a486e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:0.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} }" loading="lazy"> is used so as not to confuse the imaginary unit with current, commonly represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation>

</semantics>

</math><img src="./dc909dd0c91a68cdb9cda4ae133fa1f82c987c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }" loading="lazy">.</li></ul>

When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Capacitive reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}}</annotation>

</semantics>

</math><img src="./bbe98111b842bc759e8ca808cfc6a23d3b2b0317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle X_{C}}" loading="lazy"> and inductive reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}}</annotation>

</semantics>

</math><img src="./59f14ed86135c464841bbc6fba36cfb3997df2ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.276ex; height:2.509ex;" alt="{\displaystyle X_{L}}" loading="lazy"> contribute to the total reactance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> as follows:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {X=X_{L}+X_{C}=\omega L-{\frac {1}{\omega C}}}}</annotation>

</semantics>

</math><img src="./2b290c32d1611d25d9ae55bcb51613fece517395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.616ex; height:5.343ex;" alt="{\displaystyle {X=X_{L}+X_{C}=\omega L-{\frac {1}{\omega C}}}}" loading="lazy"></dd></dl>

where:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}}</annotation>

</semantics>

</math><img src="./59f14ed86135c464841bbc6fba36cfb3997df2ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.276ex; height:2.509ex;" alt="{\displaystyle X_{L}}" loading="lazy"> is the <a href="Inductance" title="Inductance">inductive</a> reactance, measured in ohms;</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}}</annotation>

</semantics>

</math><img src="./bbe98111b842bc759e8ca808cfc6a23d3b2b0317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle X_{C}}" loading="lazy"> is the <a href="Capacitance" title="Capacitance">capacitive</a> reactance, measured in ohms;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> is the angular frequency, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> times the frequency in <a href="Hertz" title="Hertz">Hz</a>.</li></ul>

Hence:<a href="#cite_note-Glisson-5">[5]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle X>0}</annotation>

</semantics>

</math><img src="./b09089317f0bb52f32ddcd9785bdf7c3b1baaeac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.501ex; height:1.676ex;" alt="{\displaystyle \scriptstyle X>0}" loading="lazy">, the total reactance is said to be inductive;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle X=0}</annotation>

</semantics>

</math><img src="./72fca0f827b97a232a7c709be397b6fba57f68ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.501ex; height:1.676ex;" alt="{\displaystyle \scriptstyle X=0}" loading="lazy">, then the impedance is purely resistive;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle X<0}</annotation>

</semantics>

</math><img src="./647cd6433bf7c6d44c93244e0f961228162849b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.501ex; height:1.676ex;" alt="{\displaystyle \scriptstyle X<0}" loading="lazy">, the total reactance is said to be capacitive.</li></ul>

Note however that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{L}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{L}}</annotation>

</semantics>

</math><img src="./59f14ed86135c464841bbc6fba36cfb3997df2ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.276ex; height:2.509ex;" alt="{\displaystyle X_{L}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{C}}</annotation>

</semantics>

</math><img src="./bbe98111b842bc759e8ca808cfc6a23d3b2b0317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle X_{C}}" loading="lazy"> are assumed both positive by definition, then the intermediary formula changes to a difference:<a href="#cite_note-Hughes-7">[7]</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {X=X_{L}-X_{C}=\omega L-{\frac {1}{\omega C}}}}</annotation>

</semantics>

</math><img src="./1e53a041b82181563d2847ae24cde6bfa8308ad5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.616ex; height:5.343ex;" alt="{\displaystyle {X=X_{L}-X_{C}=\omega L-{\frac {1}{\omega C}}}}" loading="lazy"></dd></dl>

but the ultimate value is the same.

<div class="mw-heading mw-heading3"><h3 id="Phase_relationship">Phase relationship</h3></div>

The phase of the voltage across a purely reactive device (i.e. with zero <a href="Parasitic_element_(electrical_networks)" class="mw-redirect" title="Parasitic element (electrical networks)">parasitic resistance</a>) lags the current by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy"> radians for a capacitive reactance and leads the current by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy"> radians for an inductive reactance. Without knowledge of both the resistance and reactance the relationship between voltage and current cannot be determined.

The origin of the different signs for capacitive and inductive reactance is the phase factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\pm \mathbf {j} {\frac {\pi }{2}}}}">

<msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\pm \mathbf {j} {\frac {\pi }{2}}}}</annotation>

</semantics>

</math><img src="./08f425d79db0dc62f584ed73df010769f077396b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.772ex; height:3.176ex;" alt="{\displaystyle e^{\pm \mathbf {j} {\frac {\pi }{2}}}}" loading="lazy"> in the impedance.

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {Z} _{C}&={1 \over \omega C}e^{-\mathbf {j} {\pi \over 2}}=\mathbf {j} \left({-{\frac {1}{\omega C}}}\right)=\mathbf {j} X_{C}\\\mathbf {Z} _{L}&=\omega Le^{\mathbf {j} {\pi \over 2}}=\mathbf {j} \omega L=\mathbf {j} X_{L}\quad \end{aligned}}}</annotation>

</semantics>

</math><img src="./a862fa55b807a3f30b99eb4cfd1ea064ee4f7005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:36.686ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {Z} _{C}&={1 \over \omega C}e^{-\mathbf {j} {\pi \over 2}}=\mathbf {j} \left({-{\frac {1}{\omega C}}}\right)=\mathbf {j} X_{C}\\\mathbf {Z} _{L}&=\omega Le^{\mathbf {j} {\pi \over 2}}=\mathbf {j} \omega L=\mathbf {j} X_{L}\quad \end{aligned}}}" loading="lazy"></dd></dl>

For a reactive component the sinusoidal voltage across the component is in quadrature (a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy"> phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power.

<ul><li><a href="Magnetic_reactance" class="mw-redirect" title="Magnetic reactance">Magnetic reactance</a></li>

<li><a href="Susceptance" class="mw-redirect" title="Susceptance">Susceptance</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Shamieh C. and McComb G., Electronics for Dummies, John Wiley & Sons, 2011.</li>

<li>Meade R., Foundations of Electronics, Cengage Learning, 2002.</li>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFYoung,_Hugh_D.Roger_A._FreedmanA._Lewis_Ford2004" class="citation book cs1">Young, Hugh D.; Roger A. Freedman; A. Lewis Ford (2004) [1949]. <a rel="nofollow" class="external text" href="https://archive.org/details/relativity00unse">Sears and Zemansky's University Physics</a> (11 ed.). <a href="San_Francisco" title="San Francisco">San Francisco</a>: <a href="Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-9179-7</bdi>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-veley01-1">^ <a href="#cite_ref-veley01_1-0">a</a> <a href="#cite_ref-veley01_1-1">b</a> <cite id="CITEREFVeley1987" class="citation book cs1">Veley, Victor F. C. (1987). <a rel="nofollow" class="external text" href="https://archive.org/details/benchtopelectron00vele">The Benchtop Electronics Reference Manual</a> (1st ed.). New York: Tab Books. pp. 229, 232.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Charles Proteus Steinmetz</a>, Frederick Bedell, <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/4763812">"Reactance"</a>, Transactions of the American Institute of Electrical Engineers, vol. 11, pp. 640–648, January–December 1894.

</li>

<li id="cite_note-Irwin-3">^ <a href="#cite_ref-Irwin_3-0">a</a> <a href="#cite_ref-Irwin_3-1">b</a> Irwin, D. (2002). Basic Engineering Circuit Analysis, page 274. New York: John Wiley & Sons, Inc.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Hayt, W.H., Kimmerly J.E. (2007). Engineering Circuit Analysis, 7th ed., McGraw-Hill, p. 388

</li>

<li id="cite_note-Glisson-5">^ <a href="#cite_ref-Glisson_5-0">a</a> <a href="#cite_ref-Glisson_5-1">b</a> Glisson, T.H. (2011). Introduction to Circuit Analysis and Design, Springer, p. 408

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Horowitz P., Hill W. (2015). <a href="The_Art_of_Electronics" title="The Art of Electronics">The Art of Electronics</a>, 3rd ed., p. 42

</li>

<li id="cite_note-Hughes-7">^ <a href="#cite_ref-Hughes_7-0">a</a> <a href="#cite_ref-Hughes_7-1">b</a> Hughes E., Hiley J., Brown K., Smith I.McK., (2012). Hughes Electrical and Electronic Technology, 11th edition, Pearson, pp. 237-241

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Robbins, A.H., Miller W. (2012). Circuit Analysis: Theory and Practice, 5th ed., Cengage Learning, pp. 554-558

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://nationalmaglab.org/magnet-academy/watch-play/interactive-tutorials/inductive-reactance/">National magnet lab, inductive reactance </a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q193972#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata719" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q193972#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata719" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4272955-5">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85111629">United States</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007563239905171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/324df0f2-0593-42b0-8384-c3dcc0fe435e">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Electrical_reactance&oldid=1304047178">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>